Question

use green's theorem to evaluate c f · dr. (check the orientation of the curve before applying the theorem.) f(x, y) = e3x x2y, e3y − xy2 c is the circle x2 y2 = 9 oriented clockwise

Answer 1

To evaluate the line integral using Green's theorem, we need to find the curl of the given vector field and confirm the orientation of the curve. Then, we can apply Green's theorem and convert the double integral to polar coordinates to evaluate the line integral. The result will give us the value of the line integral over the given curve.

Step-by-step explanation:

To evaluate the line integral using Green's theorem, we need to find the curl of the vector field F(x, y) = (e^3x * x^2y, e^3y - xy^2). The curl of a vector field can be found by taking the partial derivative of the second component with respect to x and subtracting the partial derivative of the first component with respect to y.

After finding the curl of the vector field, curl(F) = (2xy - 0, -2xy - 0), we need to find the orientation of the curve. The given circle, x^2 + y^2 = 9, is oriented clockwise.

Now we can apply Green's theorem, which states that the line integral of a vector field over a closed curve is equal to the double integral of the curl of the vector field over the region enclosed by the curve.

Using Green's theorem, we have:
∮c F · dr = ∬R curl(F) dA

Since the circle is a simple closed curve, we can use polar coordinates to evaluate the double integral. Converting the double integral to polar coordinates, we have:
∮c F · dr = ∬R curl(F) dA = ∬R (-2r^2 sinθ, -2r^2 sinθ) r dr dθ

Integrating with respect to r from 0 to 3 and with respect to θ from 0 to 2π, we can evaluate the double integral to find the value of the line integral over the given curve.

Answer 2

we obtain:

`\[ -(2)/(27) (9 - x^2)^(0.5) (101 - 65 e^(18)) e^(-9) + 4 (9 - x^2)^(1.5) \]`

This expression represents the value of the line integral
`\(\oint_C \mathbf{F} \cdot d\mathbf{r}\)`using Green's Theorem.

To evaluate the line integral
`\(\oint_C \mathbf{F} \cdot d\mathbf{r}\)` using Green's Theorem, where
`\(\mathbf{F}(x, y) = (e^(3x) x^2 y, e^(3y) - xy^2)\) and \(C\) is the circle \(x^2 + y^2 = 9\)` oriented clockwise, follow these steps:

1. Compute the Partial Derivatives:

`\[ (\partial F_1)/(\partial y) = (\partial)/(\partial y) (e^(3x) x^2 y) = e^(3x) x^2 \]`

`\[ (\partial F_2)/(\partial x) = (\partial)/(\partial x) (e^(3y) - xy^2) = -y^2 \]`

2. Apply Green's Theorem:

Green's Theorem states that

`\[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D \left( (\partial F_2)/(\partial x) - (\partial F_1)/(\partial y) \right) dA \]`

where D is the region enclosed by C. In this case,

`\[ \iint_D (-y^2 - e^(3x) x^2) dA \]`

3. Parameterize the Circle:

Since C is a circle with radius 3, we can parameterize x and y using the circle equation
`\(x^2 + y^2 = 9\).`

4. Evaluate the Integral:

We integrate over the region D bounded by the circle. The integral is

`\[ -\int_(-3)^(3) \int_(-√(9 - x^2))^(√(9 - x^2)) (y^2 + e^(3x) x^2) dy dx \]`

Note: Since the curve is oriented clockwise, we take the negative of the integral.

5. Calculate the Result

After evaluating the integral, we obtain:

`\[ -(2)/(27) (9 - x^2)^(0.5) (101 - 65 e^(18)) e^(-9) + 4 (9 - x^2)^(1.5) \]`

This expression represents the value of the line integral
`\(\oint_C \mathbf{F} \cdot d\mathbf{r}\)`using Green's Theorem.

This result is the complete step-by-step solution to the given problem using Green's Theorem.